I know chord naming and recognition is a complex dynamic, and context is important; but for now I'm only looking for names that are assigned to specific qualities of a trichord. Like "major triad" or "suspended 4th".

0,3,8 is the major triad in first inversion. 0,2,5 it an incomplete minor seventh in third inversion. 0,1,8 is likewise a major-seventh in third inversion, while 0,4,5 is in second inversion. 0,2,6 is the dominant-seventh in third inversion minus the third, while 0,3,5 is in second inversion.

I had a search based on SeuMenzes mention of quartal chords and It looks like it's called a Viannese trichord, according to wikipedia.

The problem with naming chords based on their quartal harmony, to me, is that they have to have been used enough to have a definition in themselves. Perhaps a 20th century music expert can be clearer on this, but chords as standard are stacked in 3rds. when you start using quartals you must also start considering chords stacked in 2nd's, 5ths, 6ths, 7ths, 9ths and so on and so fourth. In fact, this topic merits a question in itself!

For chords with a non-standard pattern, I believe you just get creative with the name, so for example a 1,b2 and 5 would be named be a susb2 chord. 1,b2,#5 could be an augmented susb2 chord.

I found out looking around the web that you can also get a sus2sus4 chord, (which contains 1 2 4 5). It's not a triad, but it shows how non standard chords are usually just named based on what clearly describes them.

There is not much that I can think of by now, but one could add the quartal trichord: notes stacked on perfect fourths, like C-F-Bb -- even though this could be an inversion of Bb(sus2), but with arguably different implications on overall sonority and voice-leading. The same applies to the open sus2 chord (take for example C-G-D), which has a unique sonority (harmonicizing the circle of fifths) but is also an inversion of both quartal chord and closed sus2 chord.

There are also trichords derived from modes, like the phrygian sus2 (think of it as a susb2, like in C-Db-G) and the lydian sus4 (something like sus#4, like in C-F#-G). Inversions of these trichords can be related to the locrian mode as well, and the inverted phrygian sus2 has been especially used in effect by Scriabin, becoming famous as his signature "mystic chord" (for instance, Db-G-C).